How Many Terms Are In The Geometric Series 2.1 + 10.5 + … + 820 , 312.5 2.1 + 10.5 + \ldots + 820,312.5 2.1 + 10.5 + … + 820 , 312.5 ?A. 3 B. 7 C. 9 D. 180
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Introduction
A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. In this article, we will explore the concept of a geometric series and how to determine the number of terms in a given series.
What is a Geometric Series?
A geometric series is a sequence of numbers that can be written in the form:
a, ar, ar^2, ar^3, ...
where 'a' is the first term and 'r' is the common ratio. The common ratio is the ratio of any term to its preceding term.
Example of a Geometric Series
Let's consider the geometric series: 2.1, 10.5, 42.15, 168.6, ...
In this series, the first term 'a' is 2.1 and the common ratio 'r' is 5.
How to Find the Number of Terms in a Geometric Series
To find the number of terms in a geometric series, we can use the formula:
n = (log(a_n / a) / log(r)) + 1
where 'n' is the number of terms, 'a_n' is the last term, 'a' is the first term, and 'r' is the common ratio.
Applying the Formula to the Given Series
Let's apply the formula to the given series: 2.1 + 10.5 + ... + 820,312.5.
We know that the first term 'a' is 2.1 and the last term 'a_n' is 820,312.5. We need to find the common ratio 'r'.
Finding the Common Ratio
To find the common ratio 'r', we can divide the second term by the first term:
r = 10.5 / 2.1 = 5
Applying the Formula
Now that we have the common ratio 'r', we can apply the formula to find the number of terms 'n':
n = (log(820,312.5 / 2.1) / log(5)) + 1
Calculating the Number of Terms
Using a calculator, we can calculate the number of terms 'n':
n = (log(391,500) / log(5)) + 1 n = (5.89 / 0.70) + 1 n = 8.40 + 1 n = 9.40
Since the number of terms must be an integer, we round up to the nearest whole number:
n = 10
However, we notice that the 10th term is actually 2,056,250, which is greater than the last term given in the series, 820,312.5. This means that the series must be truncated at some point.
Truncating the Series
To find the correct number of terms, we need to find the term that is closest to the last term given in the series, 820,312.5.
Finding the Correct Term
Let's calculate the 9th term:
9th term = 2.1 * (5)^8 9 term = 2.1 * 390,625 9th term = 820,312.5
This means that the 9th term is actually the last term given in the series, 820,312.5.
Conclusion
In conclusion, the number of terms in the geometric series 2.1 + 10.5 + ... + 820,312.5 is 9.
Final Answer
The final answer is 9.
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Q: What is the formula for finding the number of terms in a geometric series?
A: The formula for finding the number of terms in a geometric series is:
n = (log(a_n / a) / log(r)) + 1
where 'n' is the number of terms, 'a_n' is the last term, 'a' is the first term, and 'r' is the common ratio.
Q: How do I find the common ratio 'r' in a geometric series?
A: To find the common ratio 'r', you can divide any term by its preceding term. For example, if the series is 2.1, 10.5, 42.15, ..., you can divide the second term by the first term:
r = 10.5 / 2.1 = 5
Q: What if the series is truncated at some point? How do I find the correct number of terms?
A: If the series is truncated at some point, you need to find the term that is closest to the last term given in the series. You can do this by calculating the terms until you reach the last term given in the series.
Q: Can I use a calculator to find the number of terms in a geometric series?
A: Yes, you can use a calculator to find the number of terms in a geometric series. Simply plug in the values for 'a_n', 'a', and 'r' into the formula and calculate the result.
Q: What if the series has a negative common ratio? How do I find the number of terms?
A: If the series has a negative common ratio, you can use the same formula to find the number of terms. However, you need to be careful when calculating the terms, as the series may be alternating between positive and negative terms.
Q: Can I use the formula to find the number of terms in a finite geometric series?
A: Yes, you can use the formula to find the number of terms in a finite geometric series. Simply plug in the values for 'a_n', 'a', and 'r' into the formula and calculate the result.
Q: What if the series has a zero or infinite common ratio? How do I find the number of terms?
A: If the series has a zero or infinite common ratio, the formula will not work. In this case, you need to examine the series more closely to determine the number of terms.
Q: Can I use the formula to find the number of terms in a geometric series with a fractional common ratio?
A: Yes, you can use the formula to find the number of terms in a geometric series with a fractional common ratio. Simply plug in the values for 'a_n', 'a', and 'r' into the formula and calculate the result.
Q: What if I have a geometric series with a variable common ratio? How do I find the number of terms?
A: If you have a geometric series with a variable common ratio, you need to examine the series more closely to the number of terms. You may need to use a different formula or approach to find the number of terms.
Conclusion
In conclusion, the formula for finding the number of terms in a geometric series is:
n = (log(a_n / a) / log(r)) + 1
where 'n' is the number of terms, 'a_n' is the last term, 'a' is the first term, and 'r' is the common ratio. You can use this formula to find the number of terms in a geometric series, but you need to be careful when calculating the terms, especially if the series is truncated or has a negative common ratio.
Final Answer
The final answer is 9.